Question

If a, b, c are in AP as well as GP, then which one of the following is correct?

• A. a = b ≠ c
• B. a≠b≠c
• C. a≠b= c
• D. a = b = c

Answer 1

The correct option is D

a = b = c

a,b & c are in A.P
$\Rightarrow b-a=c-b$
$\Rightarrow 2b=c+a.....\left(1\right)$
a,b,c are in G.P also
$\Rightarrow \left(\frac{b}{a}\right)=\left(\frac{c}{b}\right)$
$\Rightarrow b^2=c\times a.....\left(2\right)$
Putting the value of b in equation (2)
${\left(\frac{(c+a)}{2}\right)}^2=4ca$
$\Rightarrow (c-a{)}^2=0$
So, c =a
Putting the value of c in equation (1), we get
$\Rightarrow 2b=a+a$
b =a
Hence, all three are equal which means a = b =c